All-wheel steering (AWS) vehicles have been widely studied in the literature to enhance stability and maneuverability. In this study, we propose a contraction theory-based AWS vehicle control strategy integrated with neural networks (NNs). Contraction theory is a powerful tool for designing controllers for nonlinear systems, ensuring the incremental exponential convergence of all system trajectories to a unique trajectory, regardless of initial conditions. However, control performance may degrade due to system uncertainties. To address this issue, NNs are employed to approximate and compensate for these system uncertainties. The contraction property is guaranteed by formulating Linear Matrix Inequalities (LMIs) to obtain the contraction metric. Furthermore, the adaptation law for the NN weights is designed using Lyapunov theory to ensure the stability of the closed-loop system. Finally, numerical simulation results are provided to validate the proposed control strategy.